 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ slarfg()

 subroutine slarfg ( integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU )

SLARFG generates an elementary reflector (Householder matrix).

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Purpose:
``` SLARFG generates a real elementary reflector H of order n, such
that

H * ( alpha ) = ( beta ),   H**T * H = I.
(   x   )   (   0  )

where alpha and beta are scalars, and x is an (n-1)-element real
vector. H is represented in the form

H = I - tau * ( 1 ) * ( 1 v**T ) ,
( v )

where tau is a real scalar and v is a real (n-1)-element
vector.

If the elements of x are all zero, then tau = 0 and H is taken to be
the unit matrix.

Otherwise  1 <= tau <= 2.```
Parameters
 [in] N ``` N is INTEGER The order of the elementary reflector.``` [in,out] ALPHA ``` ALPHA is REAL On entry, the value alpha. On exit, it is overwritten with the value beta.``` [in,out] X ``` X is REAL array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.``` [in] INCX ``` INCX is INTEGER The increment between elements of X. INCX > 0.``` [out] TAU ``` TAU is REAL The value tau.```

Definition at line 105 of file slarfg.f.

106 *
107 * -- LAPACK auxiliary routine --
108 * -- LAPACK is a software package provided by Univ. of Tennessee, --
109 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
110 *
111 * .. Scalar Arguments ..
112  INTEGER INCX, N
113  REAL ALPHA, TAU
114 * ..
115 * .. Array Arguments ..
116  REAL X( * )
117 * ..
118 *
119 * =====================================================================
120 *
121 * .. Parameters ..
122  REAL ONE, ZERO
123  parameter( one = 1.0e+0, zero = 0.0e+0 )
124 * ..
125 * .. Local Scalars ..
126  INTEGER J, KNT
127  REAL BETA, RSAFMN, SAFMIN, XNORM
128 * ..
129 * .. External Functions ..
130  REAL SLAMCH, SLAPY2, SNRM2
131  EXTERNAL slamch, slapy2, snrm2
132 * ..
133 * .. Intrinsic Functions ..
134  INTRINSIC abs, sign
135 * ..
136 * .. External Subroutines ..
137  EXTERNAL sscal
138 * ..
139 * .. Executable Statements ..
140 *
141  IF( n.LE.1 ) THEN
142  tau = zero
143  RETURN
144  END IF
145 *
146  xnorm = snrm2( n-1, x, incx )
147 *
148  IF( xnorm.EQ.zero ) THEN
149 *
150 * H = I
151 *
152  tau = zero
153  ELSE
154 *
155 * general case
156 *
157  beta = -sign( slapy2( alpha, xnorm ), alpha )
158  safmin = slamch( 'S' ) / slamch( 'E' )
159  knt = 0
160  IF( abs( beta ).LT.safmin ) THEN
161 *
162 * XNORM, BETA may be inaccurate; scale X and recompute them
163 *
164  rsafmn = one / safmin
165  10 CONTINUE
166  knt = knt + 1
167  CALL sscal( n-1, rsafmn, x, incx )
168  beta = beta*rsafmn
169  alpha = alpha*rsafmn
170  IF( (abs( beta ).LT.safmin) .AND. (knt .LT. 20) )
171  \$ GO TO 10
172 *
173 * New BETA is at most 1, at least SAFMIN
174 *
175  xnorm = snrm2( n-1, x, incx )
176  beta = -sign( slapy2( alpha, xnorm ), alpha )
177  END IF
178  tau = ( beta-alpha ) / beta
179  CALL sscal( n-1, one / ( alpha-beta ), x, incx )
180 *
181 * If ALPHA is subnormal, it may lose relative accuracy
182 *
183  DO 20 j = 1, knt
184  beta = beta*safmin
185  20 CONTINUE
186  alpha = beta
187  END IF
188 *
189  RETURN
190 *
191 * End of SLARFG
192 *
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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